Sunday, November 22, 2015

FLT65: FERMAT'S LAST THEOREM PART 4

FERMAT'S LAST THEOREM PART 4NOTE: This part contains some original mathematics. The theorems stated and proved in this section are used in the explanation that follows, but If math is not your thing, you can read the text to get the gist of it and skip over the math. The references cited will be posted with the last part of this paper.

A NOTE ON THE NATURE OF
MATHEMATICAL LOGIC AND PROOF

If FLT65 is a valid proof of
Fermat’s Last Theorem, why hasn’t it been recognized in all these years? There
are a number of reasons, and they are discussed below, but an important
underlying reason is the lack of consensus among reviewers concerning exactly what constitutes mathematical logic
and proof. We must, therefore, start by explaining what is meant by those
terms in this presentation:

Mathematical logic starts with

(1.) the inductive reasoning
process of identifying self-evident axioms underlying the logical patterns of
reality, like distinctions of subject and object, separation and unity,
enumeration and combination, difference and equivalence, extension and content,
dependent and independent variables, functional relationships, and

(2.) formalizing the fundamental
operations of substitution, addition, subtraction, multiplication and division.
Based on such axiomatic concepts and fundamental operations, the logical
structures of mathematical systems are built up by processes of deductive
reasoning.

(3.) Mathematical proof consists of using
mathematical operations and substitutions of equivalents to determine whether a
given statement or proposed theorem is consistent with accepted mathematical
logic and the axioms underlying that system.

This sounds like a boring,
rigidly structured affair that might be completed in short order. But that is,
in fact, not the case.

The history of FLT65 provides a
good example of how mathematics is not the ‘cut and dried’ subject that most
people seem to think it is. One reviewer of FLT65, a highly intelligent man
with deep interests in science and mathematics and significant math skills— but
not a professional mathematician —said, after discussing FLT65 for some time
with the author, something to the effect
that, “of all things, a mathematical
statement like a proposed proof of FLT is something we should absolutely be
able to come to an agreement about in short order, because it’s either true or
it’s not.” This reflects an erroneous belief about mathematics that was
common, even among professional mathematicians, including the great German
mathematician David Hilbert, before Kurt Gödel published his proofs of two
theorems now known as Gӧdel’s incompleteness theorems (GIT) (1929 – 1931) [6]. Gӧdel’s incompleteness
theorems are very relevant to FLT and FLT65 because they explain, in part, why
some reviewers have not understood FLT65. Additionally, after Gӧdel’s work became widely
known, many mathematicians suspected that FLT might actually be an unprovable statement.

Gӧdel’s work was a bombshell that changed mathematics forever:
It destroyed Hilbert’s dream of establishing all mathematical logic upon a finite, consistent foundation of axioms, from
which all statements could be proved either true, false or meaningless by
deductive reasoning. Finding such an all-encompassing algorithmic system was
known as the ‘entscheidungsproblem’
(decision problem) [7]. Gödel
proved, however, that no system of
mathematical logic is ever complete. This means that there can be no
complete entscheidung algorithm for
all mathematical statements. It means that there can be logically legitimate
questions posed within any given axiomatic mathematical system that are
meaningful, but unprovable within that system.

In the introduction to his
breakthrough paper, Gödel said: “As is
well known, the development of mathematics towards greater certainty has led to
the formalization of much of it such that you can complete proofs by
mechanically following a few basic rules. The most comprehensive current formal
axiomatic systems are the system of the Whitehead-Russell Principia Mathematica
on the one hand, and the Zermelo-Fraenkel axiom-system of set theory on the
other hand. These two systems are so thoroughly developed that one can
formalize in them all methods of proof currently in use in mathematics, i.e.
you can reduce the methods of constructing proofs to a few axioms and deduction
rules. Therefore, it seems plausible to [as Hilbert did][1]
that these deduction rules are sufficient to decide all mathematical questions
expressible in those systems. We will show that this is not true, but that
there are even relatively easy problems in the theory of ordinary whole numbers
that cannot be decided from existing axioms.”[8]

A mathematical statement that has not been proved is properly
called a conjecture. With a
conjecture, the question to be decided is whether the conjecture is true or
not. After Gödel, some mathematicians
believed that Fermat’s conjecture (FLT), a legitimate mathematical question,
might be an unprovable statement. This idea, however, was based on a misunderstanding of the
incompleteness theorems. The incompleteness theorems do not, as one might
suppose, imply that questions that cannot be decided within the axiomatic
system they are framed in, are necessarily ultimately unprovable. Gӧdel’s theorems did not
prove that there are undecidable questions, they proved that there are some
questions for which the mathematical system within which one is operating may
be inadequate. The system of mathematical logic within which a conjecture is
stated may have to be expanded with one or more new axioms and new definitions
and theorems before the statement can be proved to be either true or false.

In order for a reader to
understand the applicability of the “Division Algorithm” and its three
corollaries as they are used in FLT65, it
may be necessary to expand the logic of the mathematical system from which the
reviewer is operating to include the following definitions, axiomatic principle
and theorems:

Concepts
of importance in FLT65

1A Mutually
Co-Dependent Variables.

Variables are mutually co-dependent if they are related only by the form of the
algebraic expression or equation they appear in, with no designated difference
between them in terms of dependence or independence. For example, in the
equation zn - xn = yn, the variables x, y and z are mutually co-dependent. If we
designate one of them as the dependent variable and assign values to the others
so that we can set up an equation in terms of functions of the dependent
variable,the variables are no longer
mutually co-dependent. For example, if x
and y are replaced by specific
types of variables or constants, like X and Y for integer variables, or specific
values, X1 and Y1, z becomes dependent upon their values and we say the equation is of
the nth degree in z.[2]

1B. The Principle of Designation of Dependent and Independent
Variables:

Given a polynomial f(x,
y, z, … ) a function of several variables, over the field of real numbers,any of the otherwise mutually
co-dependent variables, x, y, z, …, may be designated as dependent and/or
independent, as appropriate to make the polynomial representative of meaningful
information as part of an equation or descriptive statement.

This statement expresses an
inherent fundamental feature of algebraic expressions consistent with the
axioms of a system of mathematical logic. It is a principle rather than a theorem because, while it depends upon the
validity of the axioms and fundamental operations of mathematics, it requires no multi-step proof by deductive
reasoning from those axioms. This operational principle allows us to use
polynomials to set up potentially solvable equations for application to real
problems. This principle is relevant to FLT65 and this discussion. This is so
because at several points, the designations of dependent and independent
variables are changed as needed to express functional relationships consistent
with application of the division algorithm. It was used without comment in
FLT65, and its use is made explicit here for more clarity.

2A. Parallel Polynomials:

Parallel polynomials are
polynomials of the same degree and the same algebraic form, i.e. with the
same number of terms and the same coefficients, but with different variables.
For example, a1xn +
a2xn-1 + a3xn-2 + … + an+1and a1zn
+ a2zn-1 + a3zn-2 + … + an+1
are parallel polynomials, as are Az2
+ Bzx + Cx2 and Ax2
+ Byx + Cy2.

2B. The Theorem of Parallel Factorization:

If a given polynomial,
f(z), of degree n > 1, defined over the field of real numbers, is divisible
by z –a, then the parallel polynomial f(a) is also divisible by z – a.

This theorem is proved indirectly
in FLT65, and is proved formally here for the first time, as far as the author
knows. It could also be considered to be a corollary to the division algorithm.
Note that the choice of symbols used to
represent variables in proofs is arbitrary: The validity of axioms and theorems
do not depend upon the symbols used—the use of any consistent set of symbols
would produce the same theorems. However, the symbols a, x, y, and z were
chosen to correspond to the symbols used throughout this paper to facilitate
understanding and application of the theorems to FLT65. In the generalized
two-variable algebraic polynomial forms used in the following proofs, x and z were chosen for the same reason.

Proof: Consider the generalized n-degree
polynomial:

f(z, x) =Azn +Bzn-1x
+Czn-2x2 +…+Mxn. ………………………………………………………….. 1

By the Axiom of
Unrestricted Designation of Variables, we may designate x = X1, to obtain the
single-variable polynomial

f(z) =Azn +Bzn-1X1
+Czn-2X12 + … + MX1n ………………………………………………………. 2

If we divide f(z) by z – a, by Corollary II of the Division Algorithm, the remainder,

Use of this theorem shortens the discussion of FLT and FLT65
considerably, as the reader will see in the discussions that follow. Also,
proof of the companion theorem for integers given below is important because it
removes some of the mystery expressed by reviewers concerning the application
of theorems that are true for algebraic polynomials to the subgroup involving
integers, namely integer polynomials.

2C. The Theorem of Parallel Factorization for Polynomials of
Integer Variables:

If a
given polynomial, f(Z), defined over the ring of integers is divisible by Z –a,
the parallel polynomial f(a) defined over the ring of integers is also
divisible by Z – a.

This can be inferred from the
proof of the theorem of parallel factorization for polynomials over the field
on real numbers and the fact that the integers are a subset of the set of real
numbers. Or, stated more formally: It
follows from the fact that all of the elements of the ring of integers are
contained within the set of elements of the field of real numbers. However,
importantly, it can be proved independent of the proof for polynomials over the
field of real numbers as follows:

Proof:

Given the general polynomial in two variables:

f(z,x) =Azn +Bzn-1x +Czn-2x2 +…+Mxn . ………………………………………………. 1

By the Axiom of Unrestricted Designation of Variables, we may designate x = X1a specific
integer and z = Z, an integer
variable, in the general polynomial to obtain f(Z) =AZn +BZn-1X1 +CZn-2X12
+…+MX1n. ……………………………………………. 2

And then, when we divide f(Z)
by Z – a, by Corollary II of the Division Algorithm,
the remainder is the parallel polynomial

By incorporating the operational principle, definitions and
theorems presented above into the axiomatic system of mathematical logic and
applying them to FLT, we will show how
the FLT65 demonstration that f(a),
the remainder obtained by dividing f(Z) by Z – a can never equal zero [i.e. f(a) ≠ 0], provides an unavoidable
contradiction proving Fermat’s Last Theorem.